OFFSET
0,2
COMMENTS
This is the first column of the array A082172.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..150
Valery A. Liskovets, Exact enumeration of acyclic automata, Proc. 15th Conf. "Formal Power Series and Algebr. Combin. (FPSAC'03)", 2003.
Valery A. Liskovets, Exact enumeration of acyclic deterministic automata, Discrete Appl. Math., 154, No.3 (2006), 537-551.
FORMULA
a(n) = b_3(n) where b_3(0) = 1, b_3(n) = Sum_{i=0..n-1} binomial(n, i)*(-1)^(n-i-1)*((i+2)^3 - 1)^(n-i)*b_3(i), n > 0.
MATHEMATICA
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, i] (-1)^(n - i - 1) ((i + 2)^3 - 1)^(n - i) a[i], {i, 0, n - 1}];
Table[a[n], {n, 0, 11}] (* Jean-François Alcover, Aug 29 2019 *)
PROG
(Magma)
function a(n) // a = A082160
if n eq 0 then return 1;
else return (&+[Binomial(n, j)*(-1)^(n-j-1)*((j+2)^3 - 1)^(n-j)*a(j): j in [0..n-1]]);
end if;
end function;
[a(n): n in [0..20]]; // G. C. Greubel, Jan 17 2024
(SageMath)
@CachedFunction
def a(n): # A082160
if n==0: return 1
else: return sum(binomial(n, j)*(-1)^(n-j-1)*((j+2)^3 -1)^(n-j)*a(j) for j in range(n))
[a(n) for n in range(21)] # G. C. Greubel, Jan 17 2024
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Valery A. Liskovets, Apr 09 2003
STATUS
approved